A fiber bundle (also called simply a bundle) with fiber 
is a map 
where 
is called the total space of
the fiber bundle and 
the base space of the fiber bundle. The main condition
for the map to be a fiber bundle is that every point in the
base space 
has a neighborhood 
such that 
is homeomorphic to
in a special way. Namely, if
 |
is the homeomorphism, then
 |
where the map 
means projection onto the 
component. The homeomorphisms 
which "commute with projection" are called local
trivializations for the fiber bundle 
. In other words, 
looks like the product 
(at least locally), except that the fibers 
for 
may be a bit "twisted."
A fiber bundle is the most general kind of bundle. Special cases are often described by replacing the word "fiber" with a word that describes the fiber being used, e.g., vector bundles and principal bundles.
Examples of fiber bundles include any product 
(which is a bundle over 
with fiber 
), the Möbius strip (which
is a fiber bundle over the circle with fiber
given by the unit interval [0,1]; i.e., the base space
is the circle), and 
(which is a bundle over 
with fiber 
). A special class of fiber bundle is the vector
bundle, in which the fiber is a vector
space.
Some of the properties of graphs of functions 
carry over to fiber bundles. A graph
of such a function sits in 
as 
. A graph always projects onto the base 
and is one-to-one.
A fiber bundle 
is a total space and, like 
, it has a projection 
. The preimage, 
, of any point 
is isomorphic to 
. Unlike 
, there is no canonical projection from 
to 
.
Instead, maps to 
only make sense locally on 
. Near any point 
in the base 
, there is a trivialization
of 
in which there are actual functions
from a neighborhood to 
.
These local functions can sometimes be patched together to give a (global) section 
such that the projection of 
is the identity. This is analogous to the map from a domain
of a function 
to its graph in 
by 
.
A fiber bundle also comes with a group action on the fiber. This group action represents the different ways the fiber can be viewed as equivalent. For instance, in topology, the group might be the group of homeomorphisms of the fiber. The group on a vector bundle is the group of invertible linear maps, which reflects the equivalent descriptions of a vector space using different vector bases.
Fiber bundles are not always used to generalize functions. Sometimes they are convenient descriptions of interesting manifolds. A common example in geometric topology is a torus bundle on the circle.
